On Extremal Graphs of Degree Distance Index by Using Edge-Grafting Transformations Method

Abstract

Background: Topological indices have numerous implementations in chemistry, biology and in lot of other areas. It is a real number associated to a graph, which provides information about its physical and chemical properties and their correlations. For a connected graph H, the degree distance defined as DD(H)=∑_(\h_1,h_2⊆V(H))〖(〖deg〗_H (h_1 )+〖deg〗_H (h_2 )) d_H (h_1,h_2 ) 〗, where 〖deg〗_H (h_1 ) is the degree of vertex h_1and d_H (h_1,h_2 ) is the distance between h_1and h_2in the graph H. Aim and Objective: In this article, we characterize some extremal trees with respect to degree distance index which has a lot of applications in theoretical and computational chemistry. Materials and Methods: A novel method of edge-grafting transformations is used. We discuss the behavior of DD index under four edge-grafting transformations. Results: By the help of those transformations, we derive some extremal trees under certain parameters including pendant vertices, diameter, matching and domination numbers. Some extremal trees for this graph invariant are also characterized. Conclusion: It is shown that balanced spider approaches to the smallest DD index among trees having given fixed leaves. The tree Cn,d has the smallest DD index, among the all trees of diameter d. It is also proved that the matching number and domination numbers are equal for trees having minimum DD index.

Wiener Energies of Fuzzy Graphs with Application in Total Quality Management

“…In 1972, Gutman and Trinajstic [19] introduced the Zagreb index, a degree based index, for crisp graph which is used to calculate π-electron energy related to molecular graphs. For the applications of topological indices, refer [22,23,27,28,44]. Kale and Minirani [23] defined fuzzy first Zagreb index, second Zagreb index, hyper Zagreb index and corresponding fuzzy Zagreb matrices along with fuzzy Zagreb energies.…”

“…For the applications of topological indices, refer [22,23,27,28,44]. Kale and Minirani [23] defined fuzzy first Zagreb index, second Zagreb index, hyper Zagreb index and corresponding fuzzy Zagreb matrices along with fuzzy Zagreb energies. They also presented some bounds for fuzzy Zagreb energies.…”